S ÉMINAIRE DE PROBABILITÉS (S TRASBOURG )
I SAAC M EILIJSON
On the Azéma-Yor stopping time
Séminaire de probabilités (Strasbourg), tome 17 (1983), p. 225-226
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ON THE AZEMA-YOR STOPPING TIME
by
Isaac Meilijson*
School of Mathematical Sciences
Tel-Aviv University
[1] presented
Azema and Yor
motion distributions with
stopping time
(Dubins, Root, Rost, Chacon and Walsh, and
with the
these methods
defined in terms of
limiting processes
and
explicit
S =
Let
t ~
T
=
(iii) E(T)
=
Let F
=
L(X) with E(X)
0
x
the
E(X~X ~ x)
=
y
of
proof
If
and let
ThenT Tx
a.s.,
P(X
Tx
=
0)
=
1 then r
Otherwise take
0.
=
x
Property (iii)
convergence argument
A Chacon-Walsh
on
following the
can
By continuity and
motion, it is clearly enough
of the Chacon-Walsh [2] family of
support.
to prove the
The purpose of this
Azema-Yor stopping time is
as
in [2]
special
Property (ii) becomes
stopping times.
be obtained
a
or
[3] by
a
monotone
stopping times.
stopping time (of which Dubins’ [3] is
a
special case) is
follows:
as
as
the limit of
transitions.
Embed this
Express
E(XIX >x).
F and
property (ii) given in [1] is very hard.
note is to remark that for this case the
built
=
(ii)
be the first visit to
statement for random variables X with finite
then clear.
~(x)
0 and let
a.s.
nature of Brownian
oscillatory
case
stopping time is
Var(X).
first visit to y.
The
=
Then
Property (i) is trivial.
any
The Azema-Yor
to describe:
~St > ~ (Bt)}.
inf {t
of distributions,
0} be standard Brownian motion and let
t}.
sup
decompositions
proofs.
existence
extremely simple
{Bt’
Let
or
All of
original randomized method of Skorokhod.
others), starting
are
that embeds in Brownian
Various other methods have been
mean zero.
in the literature
suggested
a
X
hitting times of
one
a
martingale with almost surely dichotomous
martingale in Brownian motion successively by first
of the two
points.
To
accomplish this, express the
226
potential function U(x) = -
(decreasing)
limit of
a
X
-x ~I
of the distribution of X
of functions defined
sequence
Uo(x) _ - ~x~,
=
is tangent to U(x).
E ~(
The
min(U (x),
anx
+
bn),
where
anx
+
the
as
as
bn
endpoints of the newly added linear piece
are
the
support of the abovementioned dichotomous transition.
If X lives
finite set, then U is
ones
these broken
through
the result is the Azema-Yor
For the sake of
procedure.
Then
0
pieces,
Consider P(X
=
one
at a
by taking
as
time, from left
0, 1
>
x.) P>
=
direct
a
i
n,
we
right,
x2... x .
1{X=x 2 }’ " ’’is1{X=x i })’
Y~,Y1, Y2,..., Yn-2’
=
X
a
martingale with
If Y is embedded in Brownian motion
it decreases.
hitting times, then, clearly, the eventual stopping time
what
to
description of the
dichotomous transition distributions, which becomes constant (and
as soon as
straight
stopping time.
completeness, here is
~
=
broken line that breaks at the
a
If the Chacon-Walsh construction is done
atoms of X.
lines the
on a
have done is
equal
to
X)
by first
embeds
L(X).
But,
exactly the Azema-Yor prescription.
REFERENCES
[1] Azéma, J. and Yor,
M.
Une solution
simple
Sem. Probab. XIII, Lecture Notes in Math.,
[2]
Chacon, R.V. and Walsh, J.B.
Dubins, L.E.
On
a
probléme
de Skorokhod.
Springer (1979).
One-dimensional
Probab. X, Lecture Notes in Math. 511,
[3]
au
potential embedding.
Springer (1976).
theorem of Skorokhod.
Ann.Math.Statist., 39 (1968),
2094-2097.
*
This work
was
performed during
University, Amsterdam, 1980/1.
Sem.
the author’s visit to the Free